Singular Value Decomposition (SVD)

Any m × n matrix A can be factored as A = U Σ Vᵀ, where U (m × m) and V (n × n) are orthogonal matrices whose columns are the left and right singular vectors, and Σ (m × n) is a diagonal matrix of singular values σ₁ ≥ σ₂ ≥ ... ≥ σᵣ > 0. This decomposition exists for every matrix, unlike eigendecomposition which requires square, diagonalizable matrices.

The condition number of a matrix is the ratio of its largest to smallest singular value: κ(A) = σₘₐₓ / σₘᵢₙ. A high condition number indicates an ill-conditioned system where small input perturbations cause large solution errors. In DPD coefficient estimation, ill-conditioned correlation matrices arise from highly correlated input signals, making the Normal Equation solution numerically unstable. SVD-based pseudoinversion bypasses this by truncating near-zero singular values.

SVD reveals the fundamental subspaces of a matrix:

• The first r columns of U span the column space (range)
• The last m − r columns of U span the left nullspace
• The first r columns of V span the row space
• The last n − r columns of V span the nullspace The singular values represent the semi-axes lengths of the hyperellipsoid formed by mapping the unit sphere through A.

The Moore-Penrose pseudoinverse is computed as A⁺ = V Σ⁺ Uᵀ, where Σ⁺ is formed by taking the reciprocal of each non-zero singular value and transposing. For ill-conditioned systems, a truncated SVD sets σᵢ⁺ = 0 for σᵢ below a threshold, effectively regularizing the solution. This provides the minimum-norm least-squares solution even when AᵀA is singular, making it superior to the Normal Equation for DPD parameter extraction.

The Eckart-Young-Mirsky theorem states that the optimal rank-k approximation to A in both Frobenius and spectral norms is obtained by retaining only the k largest singular values and their corresponding vectors: Aₖ = Uₖ Σₖ Vₖᵀ. This property is exploited in model order reduction for DPD, where a high-order memory polynomial can be compressed by discarding singular vectors corresponding to negligible singular values, reducing computational complexity without sacrificing linearization performance.

The singular values of A are the square roots of the eigenvalues of both AᵀA and AAᵀ. The right singular vectors V are the eigenvectors of AᵀA, and the left singular vectors U are the eigenvectors of AAᵀ. This relationship explains why SVD is preferred over direct eigendecomposition of AᵀA for solving least-squares problems: forming AᵀA squares the condition number, doubling the loss of numerical precision.